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 Details of the OPTEX Procedure

## Example 24.2: Comparing DETMAX Algorithm to Sequential Algorithm

 See OPTEX4 in the SAS/QC Sample Library

An automotive engineer wants to fit a quadratic model to fuel consumption data in order to find the values of the control variables that minimize fuel consumption (refer to Vance 1986). The three control variables and their possible settings are shown in the following table:

 Variable Values AF 15 16 17 18 EGR 0.020 0.177 0.377 0.566 0.921 1.117 SA 10 16 22 28 34 40 46 52

Rather than run all 192 (4×6×8) combinations of these factors, the engineer would like to see whether the total number of runs can be reduced to 50 in an optimal fashion.

Since the factors have different numbers of levels, you can use the PLAN procedure (refer to the SAS/STAT User's Guide) to generate the full factorial set to serve as a candidate data set for the OPTEX procedure.

```   proc plan;
factors af=4 ordered egr=6 ordered sa=8 ordered
/ noprint;
output out=a
af  nvals=(15,16,17,18)
egr nvals=(.020,.177,.377,.566,.921,1.117)
sa  nvals=(10,16,22,28,34,40,46,52);
run;
```

The DETMAX algorithm of Mitchell (1974a) is very commonly used for computer-generated optimal design. Although it is not the default search method for the OPTEX procedure, you can specify that it be used with the METHOD=DETMAX option in the GENERATE statement. For example, the following statements produce Output 24.2.1.

```   proc optex data=a seed=61552;
model af|egr|sa@2 af*af egr*egr sa*sa;
generate n=50 method=detmax;
run;
```

Output 24.2.1: Efficiencies with DETMAX Algorithm

 The OPTEX Procedure

 Design Number D-Efficiency A-Efficiency G-Efficiency Average PredictionStandard Error 1 46.4922 24.8987 95.2281 0.4202 2 46.4864 24.8562 95.5744 0.4205 3 46.4797 24.8830 95.3137 0.4203 4 46.4635 25.6461 94.8125 0.4175 5 46.4495 24.5376 95.5559 0.4237 6 46.4459 25.0749 94.8536 0.4197 7 46.4428 24.5111 95.3704 0.4240 8 46.4333 25.0321 95.1371 0.4199 9 46.4333 25.0321 95.1371 0.4199 10 46.4333 25.0321 95.1371 0.4199

The DETMAX search method can require considerable run time. For comparison, you can use the METHOD=SEQUENTIAL option in the GENERATE statement, as shown in the following statements, which produce Output 24.2.2.
```   proc optex data=a seed=33805;
model af|egr|sa@2 af*af egr*egr sa*sa;
generate n=50 method=sequential;
run;
```

Output 24.2.2: Efficiencies with Sequential Algorithm

 The OPTEX Procedure

 Design Number D-Efficiency A-Efficiency G-Efficiency Average PredictionStandard Error 1 46.4009 25.0472 93.8673 0.4200

In a fraction of the run time required by DETMAX, the sequential algorithm finds a design with a relative D-efficiency of 46.4009 / 46.4922 = 99.8% compared to the best design found by the DETMAX procedure and with better A-efficiency. As this demonstrates, if absolute D-optimality is not required, a faster, simpler search may be sufficient.

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